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<div class="titlepage"><div><div><h4 class="title">
<a name="math_toolkit.dist_ref.dists.bernoulli_dist"></a><a class="link" href="bernoulli_dist.html" title="Bernoulli Distribution">Bernoulli
        Distribution</a>
</h4></div></div></div>
<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">distributions</span><span class="special">/</span><span class="identifier">bernoulli</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span></pre>
<pre class="programlisting"><span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">math</span><span class="special">{</span>
 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">RealType</span> <span class="special">=</span> <span class="keyword">double</span><span class="special">,</span>
           <span class="keyword">class</span> <a class="link" href="../../../policy.html" title="Chapter 22. Policies: Controlling Precision, Error Handling etc">Policy</a>   <span class="special">=</span> <a class="link" href="../../pol_ref/pol_ref_ref.html" title="Policy Class Reference">policies::policy&lt;&gt;</a> <span class="special">&gt;</span>
 <span class="keyword">class</span> <span class="identifier">bernoulli_distribution</span><span class="special">;</span>

 <span class="keyword">typedef</span> <span class="identifier">bernoulli_distribution</span><span class="special">&lt;&gt;</span> <span class="identifier">bernoulli</span><span class="special">;</span>

 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">RealType</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../../policy.html" title="Chapter 22. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
 <span class="keyword">class</span> <span class="identifier">bernoulli_distribution</span>
 <span class="special">{</span>
 <span class="keyword">public</span><span class="special">:</span>
    <span class="keyword">typedef</span> <span class="identifier">RealType</span>  <span class="identifier">value_type</span><span class="special">;</span>
    <span class="keyword">typedef</span> <span class="identifier">Policy</span>    <span class="identifier">policy_type</span><span class="special">;</span>

    <span class="identifier">bernoulli_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">p</span><span class="special">);</span> <span class="comment">// Constructor.</span>
    <span class="comment">// Accessor function.</span>
    <span class="identifier">RealType</span> <span class="identifier">success_fraction</span><span class="special">()</span> <span class="keyword">const</span>
    <span class="comment">// Probability of success (as a fraction).</span>
 <span class="special">};</span>
<span class="special">}}</span> <span class="comment">// namespaces</span>
</pre>
<p>
          The Bernoulli distribution is a discrete distribution of the outcome of
          a single trial with only two results, 0 (failure) or 1 (success), with
          a probability of success p.
        </p>
<p>
          The Bernoulli distribution is the simplest building block on which other
          discrete distributions of sequences of independent Bernoulli trials can
          be based.
        </p>
<p>
          The Bernoulli is the binomial distribution (k = 1, p) with only one trial.
        </p>
<p>
          <a href="http://en.wikipedia.org/wiki/Probability_density_function" target="_top">probability
          density function pdf</a>
        </p>
<div class="blockquote"><blockquote class="blockquote"><p>
            <span class="serif_italic">f(0) = 1 - p, f(1) = p</span>
          </p></blockquote></div>
<p>
          <a href="http://en.wikipedia.org/wiki/Cumulative_Distribution_Function" target="_top">Cumulative
          distribution function</a>
        </p>
<div class="blockquote"><blockquote class="blockquote"><p>
            <span class="serif_italic">D(k) = if (k == 0) 1 - p else 1</span>
          </p></blockquote></div>
<p>
          The following graph illustrates how the <a href="http://en.wikipedia.org/wiki/Probability_density_function" target="_top">probability
          density function pdf</a> varies with the outcome of the single trial:
        </p>
<div class="blockquote"><blockquote class="blockquote"><p>
            <span class="inlinemediaobject"><img src="../../../../graphs/bernoulli_pdf.svg" align="middle"></span>

          </p></blockquote></div>
<p>
          and the <a href="http://en.wikipedia.org/wiki/Cumulative_Distribution_Function" target="_top">Cumulative
          distribution function</a>
        </p>
<div class="blockquote"><blockquote class="blockquote"><p>
            <span class="inlinemediaobject"><img src="../../../../graphs/bernoulli_cdf.svg" align="middle"></span>

          </p></blockquote></div>
<h5>
<a name="math_toolkit.dist_ref.dists.bernoulli_dist.h0"></a>
          <span class="phrase"><a name="math_toolkit.dist_ref.dists.bernoulli_dist.member_functions"></a></span><a class="link" href="bernoulli_dist.html#math_toolkit.dist_ref.dists.bernoulli_dist.member_functions">Member
          Functions</a>
        </h5>
<pre class="programlisting"><span class="identifier">bernoulli_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">p</span><span class="special">);</span>
</pre>
<p>
          Constructs a <a href="http://en.wikipedia.org/wiki/bernoulli_distribution" target="_top">bernoulli
          distribution</a> with success_fraction <span class="emphasis"><em>p</em></span>.
        </p>
<pre class="programlisting"><span class="identifier">RealType</span> <span class="identifier">success_fraction</span><span class="special">()</span> <span class="keyword">const</span>
</pre>
<p>
          Returns the <span class="emphasis"><em>success_fraction</em></span> parameter of this distribution.
        </p>
<h5>
<a name="math_toolkit.dist_ref.dists.bernoulli_dist.h1"></a>
          <span class="phrase"><a name="math_toolkit.dist_ref.dists.bernoulli_dist.non_member_accessors"></a></span><a class="link" href="bernoulli_dist.html#math_toolkit.dist_ref.dists.bernoulli_dist.non_member_accessors">Non-member
          Accessors</a>
        </h5>
<p>
          All the <a class="link" href="../nmp.html" title="Non-Member Properties">usual non-member accessor
          functions</a> that are generic to all distributions are supported:
          <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.cdf">Cumulative Distribution Function</a>,
          <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability Density Function</a>,
          <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.quantile">Quantile</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.hazard">Hazard Function</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.chf">Cumulative Hazard Function</a>,
          <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.mean">mean</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.median">median</a>,
          <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.mode">mode</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.variance">variance</a>,
          <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.sd">standard deviation</a>,
          <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.skewness">skewness</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.kurtosis">kurtosis</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.kurtosis_excess">kurtosis_excess</a>,
          <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.range">range</a> and <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.support">support</a>.
        </p>
<p>
          The domain of the random variable is 0 and 1, and the useful supported
          range is only 0 or 1.
        </p>
<p>
          Outside this range, functions are undefined, or may throw domain_error
          exception and make an error message available.
        </p>
<h5>
<a name="math_toolkit.dist_ref.dists.bernoulli_dist.h2"></a>
          <span class="phrase"><a name="math_toolkit.dist_ref.dists.bernoulli_dist.accuracy"></a></span><a class="link" href="bernoulli_dist.html#math_toolkit.dist_ref.dists.bernoulli_dist.accuracy">Accuracy</a>
        </h5>
<p>
          The Bernoulli distribution is implemented with simple arithmetic operators
          and so should have errors within an epsilon or two.
        </p>
<h5>
<a name="math_toolkit.dist_ref.dists.bernoulli_dist.h3"></a>
          <span class="phrase"><a name="math_toolkit.dist_ref.dists.bernoulli_dist.implementation"></a></span><a class="link" href="bernoulli_dist.html#math_toolkit.dist_ref.dists.bernoulli_dist.implementation">Implementation</a>
        </h5>
<p>
          In the following table <span class="emphasis"><em>p</em></span> is the probability of success
          and <span class="emphasis"><em>q = 1-p</em></span>. <span class="emphasis"><em>k</em></span> is the random
          variate, either 0 or 1.
        </p>
<div class="note"><table border="0" summary="Note">
<tr>
<td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../../../doc/src/images/note.png"></td>
<th align="left">Note</th>
</tr>
<tr><td align="left" valign="top">
<p>
            The Bernoulli distribution is implemented here as a <span class="emphasis"><em>strict
            discrete</em></span> distribution. If a generalised version, allowing
            k to be any real, is required then the binomial distribution with a single
            trial should be used, for example:
          </p>
<p>
            <code class="computeroutput"><span class="identifier">binomial_distribution</span><span class="special">(</span><span class="number">1</span><span class="special">,</span>
            <span class="number">0.25</span><span class="special">)</span></code>
          </p>
</td></tr>
</table></div>
<div class="informaltable"><table class="table">
<colgroup>
<col>
<col>
</colgroup>
<thead><tr>
<th>
                  <p>
                    Function
                  </p>
                </th>
<th>
                  <p>
                    Implementation Notes
                  </p>
                </th>
</tr></thead>
<tbody>
<tr>
<td>
                  <p>
                    Supported range
                  </p>
                </td>
<td>
                  <p>
                    {0, 1}
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    pdf
                  </p>
                </td>
<td>
                  <p>
                    Using the relation: pdf = 1 - p for k = 0, else p
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    cdf
                  </p>
                </td>
<td>
                  <p>
                    Using the relation: cdf = 1 - p for k = 0, else 1
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    cdf complement
                  </p>
                </td>
<td>
                  <p>
                    q = 1 - p
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    quantile
                  </p>
                </td>
<td>
                  <p>
                    if x &lt;= (1-p) 0 else 1
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    quantile from the complement
                  </p>
                </td>
<td>
                  <p>
                    if x &lt;= (1-p) 1 else 0
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    mean
                  </p>
                </td>
<td>
                  <p>
                    p
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    variance
                  </p>
                </td>
<td>
                  <p>
                    p * (1 - p)
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    mode
                  </p>
                </td>
<td>
                  <p>
                    if (p &lt; 0.5) 0 else 1
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    skewness
                  </p>
                </td>
<td>
                  <p>
                    (1 - 2 * p) / sqrt(p * q)
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    kurtosis
                  </p>
                </td>
<td>
                  <p>
                    6 * p * p - 6 * p +1/ p * q
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    kurtosis excess
                  </p>
                </td>
<td>
                  <p>
                    kurtosis -3
                  </p>
                </td>
</tr>
</tbody>
</table></div>
<h5>
<a name="math_toolkit.dist_ref.dists.bernoulli_dist.h4"></a>
          <span class="phrase"><a name="math_toolkit.dist_ref.dists.bernoulli_dist.references"></a></span><a class="link" href="bernoulli_dist.html#math_toolkit.dist_ref.dists.bernoulli_dist.references">References</a>
        </h5>
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
<li class="listitem">
              <a href="http://en.wikipedia.org/wiki/Bernoulli_distribution" target="_top">Wikipedia
              Bernoulli distribution</a>
            </li>
<li class="listitem">
              <a href="http://mathworld.wolfram.com/BernoulliDistribution.html" target="_top">Weisstein,
              Eric W. "Bernoulli Distribution." From MathWorld--A Wolfram
              Web Resource.</a>
            </li>
</ul></div>
</div>
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      Walker and Xiaogang Zhang<p>
        Distributed under the Boost Software License, Version 1.0. (See accompanying
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